Chapter 1

Question 1

Euclidean distance on the real line

Answer 1

The function d: R x R -> [0, inf) defined by
d(x,y) = |x-y|,
for all x, y e R.

Question 2

Properties of the Euclidean distance

Answer 2

For any x,y,z e R

1) |x-y| >= 0 and |x-y| =0 iff x=y
(2) | x-y | = |y-x| (Symmetry
(3) |x-y| =< |x-z| +|z-y| (Triangle inequality)

Question 3

Open interval centred at x0

Answer 3

Given x0 e , an open interval centred at x0 is a set of real numbers of the form (x0-r, x0+r) for some r>0.
(x0-r, x0+r) = {x e R: |x - x0| < r}

Question 4

Open set

Answer 4

A subset U C_ R is a open set if:
For every x e U, there exists e>0 such that (x-e, x+e) C_ U

A set U of real numbers is an open set if it has the property that for every point x in U there is an open interval centred at that point that is contained in U.

Question 5

Closed set

Answer 5

A subset F C_ R is a closed set if the complement of F in R (F^c) is an open set.

Question 6

Bounded set

Answer 6

A subset Y C_ R is a bounded set if there exists K e R such that |x| < K for all x E Y.

Question 7

The union of open sets

Answer 7

The union of an arbitrary family of open sets is an open set

Question 8

The intersection of open sets

Answer 8

The intersection of a finite number of open sets is an open set.

Question 9

The union of closed sets

Answer 9

The union of a finite number of closed sets is a closed set

Question 10

The intersection of closed sets

Answer 10

The intersection of a arbitrary family of closed sets is a closed set